Let \(D\)  be a division ring with center \(F\)  and \(D^*\)  be the multiplicative group of \(D\) . The subgroup structure of \(D^*\)  is one of subjects which attract the attention of many authors for last time. A particular interest is that to study maximal subgroups of \(D^*\) , see for example, [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. In this paper we replace \(D^*\)  by its arbitrary subnormal subgroup \(G\)  and we study the subgroup structure of \(G\) . Recall that in [1]}, [3]}, [6]}, Akbari et al. and Mahdavi-Hezavehi study maximal subgroups of \(D^*\)  and many nice properties of such subgroups were obtained. In the present paper, studying maximal subgroups of \(G\) , we get in many cases the similar results for these subgroups as the results obtained in [1]}, [3]}, [6]} ... for maximal subgroups of \(D^*\) . Other problem we study is the problem of the existence of noncyclic free groups in a subnormal subgroups of a division ring. This problem plays an important role for the understanding of the structure of division rings. For more information we refer to the works [13]}-[14]}, [15]}-[16]}, and [17]}. Section 4 is devoted to this problem and the new result we get in Theorem REF  gives the affirmative answer to Conjecture 2 from [14]} for the case of locally finite division rings.
